The Lambda Calculus for Absolute Dummies (2012)

Audio: "This is the basics of lambda calculus right here. You’ve got the little lambda function, a lambda character and that’s indicates the beginning of a function definition. you say this function takes a varia..uh argument called x. The period denotes the beginning of the body of the function, and then whatever comes after the period is what is returned by the function. So this is an identify function here. Anything you give it, it returns back the same value. You have arguments, well ok you have variables, and you have functions. That's all you have in lambda calculus. There is nothing else, but variables and functions. "

Visual: There is a title to the slide, "λ-calculus". There is an image on the center of the slide with λx.x, and there are these words with arrows going to λx.x :

"punctuation", with two arrows coming out of it, going λ and . "argument (variable)" with an arrow to the first x "body (λ-expression)" with an arrow to the second x.

This person would fail the first definition-proof based math exam at Harvard. Let's say, the question is "What is a function?"

You can't say, well, uh, you have your f there, so basically, that's the function, and you have these arrows, and the x between the arrows, that's uh, your variable. f takes that, and uh, returns whatever's after the equal. So that's basically a function.

And I'd draw an image of f(x) = y, and have arrows to ()= saying punctuation, to f, "function", and x,y "variables".

Sloppy thinking like this is what resulted in inconsistencies in calculus, until people got rigorous and started giving good definitions of what exactly they're talking about. Here's a standard definition:

http://en.wikipedia.org/wiki/Function_(mathematics)#Definiti...

Common Intuition is not always a good guide for mathematics or computer science. The world is not as it seems. You have things like the Banach Tarski paradox, or Brouwer's fixed-point theorem which may be counterintuitive but once things clearly defined, makes more sense.

This article, and your comment, reminded me of 'Math Class (Imagined by Kids)' http://www.youtube.com/watch?v=KdxEAt91D7k (2 minutes long)

  def teach(something):
      lesson = None
      // Aye, there's the rub...
      if knowsHowToTeach():
          lesson = createLessonForSomething(something)
      else:
          lesson = teach(something)
      return lesson

It's elegant, but elitist, because it's unnecessarily difficult and hostile.

[1]: http://mathoverflow.net/questions/152352/is-euclid-dead